#### 1. Abstract

During a snowfall, the snow crystals accumulate on the ground and gradually form a complex porous medium constituted of air, water vapor, ice and sometimes liquid water. This ground-lying snow transforms with time, depending on the physical parameters of the environment. This process, called metamorphism, can be divided into three main types of metamorphisms: the wet snow metamorphism, the isothermal metamorphism, and the temperature gradient (TG) metamorphism. In polar or mountainous conditions, these processes are followed by compaction of snow into ice under the load of the upper snow layers. Although the general effects of theses various types of metamorphism are roughly well-known, the physical mechanisms that lead to these transformations are not perfectly well understood. A better microscale simulation of these mechanisms would help to understand how snow changes its microstructure depending on the applied physical conditions and has major impacts on the studies of snow behavior and properties at larger scale.

The main purpose of this project is to provide efficient computational tools to study the snow metamorphism from 3D images of real snow microstructures acquired using X-ray tomography techniques. In particular, our work will focus on the development of 3D image-based numerical models that can simulate the shape evolution of the snow microstructure during its metamorphism. It will be completed by other tools designed to extract physical properties from the computed microstructures.

The resolution of these problems requires a strong interaction between three disciplines: Snow Physics, Applied Mathematics and Computer Science, which justifies the collaboration between the CEN, the LAMA and the LIRIS respectively. To achieve our objectives, three key points are required: obtaining 3D images of snow microstructures, representing the digital objects with adapted data structures in order to perform numerical measurements, and evolving the digital objects according to physical laws. Last but not least, experimental validations of the proposed tools and models are required. The project is composed of the four following tasks:

- Snow metamorphism modelling
- HP digital toolbox for volumetric and surface analysis
- Dynamic digital structures for evolving regions
- Physical rendering for snow radiative transfer

This project has several impacts in Snow Physics, Applied Mathematics and Computer Science. The main results will be made available to the scientific community using the DGtal open source library.

#### 2. Context, positioning and objectives of the project

##### 2.1 Context of the project

The origin of the project comes from discussion between computer scientists and researchers from CEN (Météo-France) working on snow microstructure. During a snowfall, the snow crystals accumulate on the ground and gradually form a complex porous medium constituted of air, water vapour, ice and sometimes liquid water. This ground-lying snow transforms with time, depending on the physical parameters of the environment. This process, called metamorphism, can be divided into three main types of metamorphisms: the wet snow metamorphism, the isothermal metamorphism, and the temperature gradient (TG) metamorphism. In polar or mountainous conditions, these processes are followed by compaction of snow into firn and ice under the load of the upper snow layers. Although the general effects of theses various types of metamorphism are roughly well-known, the physical mechanisms that lead to these transformations are not perfectly well understood. Furthermore, physical and mechanical properties of snow are strongly depending on its microstructure: studying snow metamorphism offers interesting outcomes to forecast the evolution of important parameters of the snowpack (e.g. density, thermal conductivity, albedo…) with time. This knowledge is of particular interest as it is used in large-scale snowpack models like CROCUS (Brun et al. 1992) or SNOWPACK (Lehning et al. 2002), which are devoted to highly societal purposes such as operational avalanche forecasting or climate studies.

To improve the current knowledge of snow metamorphism, realistic simulations of snow morphological changes with time and comparisons to quantitative measurements of snow microstructure are mandatory. New techniques such as confocal imaging or microtomography are becoming easily available and produce precise three-dimensional (3D) information from the inside of the investigated samples, without damaging the sample. By using appropriate tools, it is then possible to extract a lot of information from these original images, and to simulate specific physical mechanisms in 3D. This process leads to very interesting challenges in Computer Sciences and Applied Mathematics:

- From physical equations and experimental behaviors expressed by CEN researchers, how to model snow metamorphism with mathematical and numerical models?
- How to handle high resolution 3D images (up to 2048^3) and perform reliable measurements on objects and regions contained in these images?
- How to mix up the two previous points in order to compute physical driven deformations on regions in huge 3D images?

These challenges give rise to several bottlenecks in Applied Mathematics since non-linear partial differential equations may appear on interface structures. In this context the expertise of the LAMA-EDPs2 team in both the analysis of partial differential equations and numerical analysis is of crucial importance. Moreover, the promising interactions between the EDPs2 team and the members of the digital geometry community (LAMA and LIRIS partners) will give us the possibility to develop efficient mixed approaches which combine the advantages of discrete and continuous models.

For Computer Sciences, bottlenecks lie in the field of Digital Geometry in which both the LIRIS-M2DisCo and LAMA-LIMD teams are recognized as specialists. Indeed, Digital Geometry can be simply characterised as a set of definitions, theorems and algorithmic tools that deal with the geometric and topological properties of subsets of digital pictures. A more generic definition considers the analysis of data structured on regular lattices (refer to (Klette & Rosenfeld 2004) and (Coeurjolly et al. 2007) for an overview of the domain). Even if the domain emerged during the second half of the 20th century with the birth of computer graphics and digital image processing, many links have been demonstrated between Digital Geometry results and fundamental theorems in mathematics (arithmetic, geometry, topology…), discrete mathematics (word theory, combinatorics, graph theory…) or computer science (algorithmic, image processing…). It aims at modelling and analysing continuous objects or phenomena with the help of a finite number of discrete data. Since we consider objects on regular grids (finite subset of grid points), we usually represent these objects with integer numbers. Hence, thanks to the discrete nature of the objects, algorithms in Digital Geometry are generally sped-up by considering properties from arithmetic or computational geometry.

Another challenge in Computer Sciences is to investigate the role of physical based rendering techniques developed for realistic image synthesis in order to evaluate radiance parameters of snow samples. This last task matches with skills and activities of the R3AM team (Realistic Rendering for Mobile Augmented Reality) at LIRIS.

*To be continued…*